Cruz Roberto, Monsalve Juan, Rada Juan
Instituto de Matemáticas, Universidad de Antioquia, Medellín, Colombia.
Heliyon. 2022 Nov 24;8(11):e11874. doi: 10.1016/j.heliyon.2022.e11874. eCollection 2022 Nov.
We assume that is a directed graph with vertex set and arc set . A VDB topological index of is defined as where and denote the outdegree and indegree of vertices and , respectively, and is a bivariate symmetric function defined on nonnegative real numbers. Let be the general adjacency matrix defined as if , and 0 otherwise. The energy of with respect to a VDB index is defined as , where are the singular values of the matrix . We will show that in case is the Randić index, the spectral norm of is equal to 1, and rank of is equal to rank of the adjacency matrix of . Immediately after, we illustrate by means of examples, that these properties do not hold for most well-known VDB topological indices. Taking advantage of nice properties the Randić matrix has, we derive new upper and lower bounds for the Randić energy in digraphs. Some of these generalize known results for the Randić energy of graphs. Also, we deduce a new upper bound for the Randić energy of graphs in terms of rank, concretely, we show that for all graphs , and equality holds if and only if is a disjoint union of complete bipartite graphs.
我们假设(G)是一个具有顶点集(V(G))和弧集(A(G))的有向图。(G)的一个VDB拓扑指数(I(G))定义为(I(G)=\sum_{u,v\in V(G)}f(d^+(u),d^-(v)))，其中(d^+(u))和(d^-(v))分别表示顶点(u)和(v)的出度和入度，并且(f)是一个定义在非负实数上的二元对称函数。设(A(G))是一般邻接矩阵，定义为如果((u,v)\in A(G))，则(a_{uv}=1)，否则(a_{uv}=0)。关于VDB指数(I(G))的(G)的能量定义为(E_{I(G)}(G)=\sum_{i = 1}^n\sigma_i)，其中(\sigma_i)是矩阵(A(G))的奇异值。我们将证明，当(I(G))是兰迪奇指数时，(A(G))的谱范数等于(1)，并且(A(G))的秩等于(G)的邻接矩阵的秩。紧接着，我们通过例子说明，这些性质对于大多数著名的VDB拓扑指数并不成立。利用兰迪奇矩阵所具有的良好性质，我们推导出有向图中兰迪奇能量的新的上下界。其中一些推广了图的兰迪奇能量的已知结果。此外，我们根据秩推导出图的兰迪奇能量的一个新的上界，具体地说，我们证明对于所有的图(G)，(RE(G)\leq\sqrt{2m(G)r(G)})，当且仅当(G)是完全二部图的不相交并时等号成立。
